THÈSE DE DOCTORAT DE L'UNIVERSITÉ PARIS 6 Spécialité ...

52 Exact minimax estimation in sup-norm for anisotropic Hölder classesLemma 3.4. Let Q : R d → R be a function such that ‖Q‖ 2 2 = ∫ Q 2 < ∞, ∆ be aR dcompact set ∆ = ∏ di=1 ∆ i with ∆ i intervals of [0, +∞) of length T i > 0 and W be thestandard Brownian sheet on ∆. Let h 1 , . . . , h d be arbitrary positive numbers and we writeh = ∏ di=1 h i. We consider the gaussian process defined for t = (t 1 , . . . , t d ) ∈ ∆:(1u1 − tX t = √ 1Q , . . . ,h1 · · · h d∫R u )d − t ddW u , (3.11)h d dwith u = (u 1 , . . . , u d ). Let (α 1 , . . . , α d ) ∈ (0, ∞) d and let α be the number such that1/α = ∑ di=1 1/α i. Let T = ∏ di=1 T i. We suppose that there exists 0 < c 1 < ∞ such that,for t ∈ [−1, 1] d ,∫(2d∑(Q(t + u) − Q(u)) 2 du ≤ c 1 |t i | i) α . (3.12)R dThen there exists a constant c 2 > 0, such that for b ≥ c 2 /| log h| 1/2 and h small enough,[]) ()P sup |X t | ≥ b ≤ N(h) exp(− b2c 2 bexp, (3.13)t∈∆2‖Q‖ 2 2 ‖Q‖ 2 2| log h| 1/2where c 2 = c 3 (c 4 + 1/ √ α), c 3 and c 4 do not depend on h 1 , . . . , h d , T and α, P denotes thedistribution of {X t , t ∈ ∆} andN(h) = 2d∏i=1h 1i=1(Tih i(c1 d| log h| 1/2) 1/α i+ 1).Note that if the h i /T i → 0, then for the h i /T i small enoughN(h) ≤ 2 d+1 T h(c1 d| log h| 1/2) 1/α.This lemma is close to various results on the supremum of Gaussian processes (seeAdler (1990), Lifshits (1995), Piterbarg (1996)). The closest result is Theorem 8-1 ofPiterbarg (1996) which, however, cannot be used directly since there is no explicit expressionfor the constants that in our case depend on h and T and may tend to 0 or ∞. Alsothe explicit dependence of the constants on α is given here. This can be useful for thepurpose of adaptive estimation.Proof. Let λ > 0 and N 1 (λ, S) be the minimal number of hyperrectangles with edges of( )λ 1/α1,. ( 1/αdlength h 1 c 1 d . .λ ,hdc 1 d)that cover a set S ⊂ ∆. We haveN 1 (λ, ∆) ≤d∏i=1([ ( ) ] )1/αiT i c1 d+ 1 ,h i λ